We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and provides probabilistic insight into the phenomenon. In the case of multi-class networks, our results generalise those of Bonald and Proutière (2002), (2003).

Consider a continuous-time Markov chain evolving in a random environment. We study certain forms of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. Moreover, we obtain necessary and sufficient conditions for a product-form stationary distribution. A number of examples that illustrate the applicability of our results in queueing and population growth models are also included.

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.

In this paper we consider a network of queues with batch services, customer coalescence and state-dependent signaling. That is, customers are served in batches at each node, and coalesce into a single unit upon service completion. There are signals circulating in the network and, when a signal arrives at a node, a batch of customers is either deleted or triggered to move as a single unit within the network. The transition rates for both customers and signals are quite general and can depend on the state of the whole system. We show that this network possesses a product form solution. The existence of a steady state distribution is also discussed. This result generalizes some recent results of Henderson et al. (1994), as well as those of Chao et al. (1996).

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.

We consider a migration process whose singleton process is a time-dependent Markov replacement process. For the singleton process, which may be treated as either open or closed, we study the limiting distribution, the distribution of the time to replacement and related quantities. For a replacement process in equilibrium we obtain a version of Little's law and we provide conditions for reversibility. For the resulting linear population process we characterize exponential ergodicity for two types of environmental behaviour, i.e. either convergent or cyclic, and finally for large population sizes a diffusion approximation analysis is provided.

Queueing systems with non-exponential service distributions and a capacity restriction on the total residual service requirement (workload) are studied. Closed-form expressions of geometric form in the LCFS case and recursive representations in the FCFS case are obtained for the steady state workload distribution. The proof is of interest in itself and directly leads to extensions.

Many authors have discussed the equivalence of partial balance and insensitivity in in stochastic processes. When speeds are introduced into a stochastic process there arises a difficulty in proving the necessity of partial balance for insensitivity. Previous authors have overcome this difficulty by assuming that a process has the property of instantaneous attention. This property enforces the requirement that no lifetime can be created in a state in which that lifetime has zero speed.

In this paper it is shown that for processes with a finite state space it is unnecessary to make this assumption provided the notion of partial balance is slightly changed. Thus we give a criterion, analogous to partial balance, which is necessary and sufficient for insensitivity even in processes which do not possess the property of instantaneous attention. When a process does have instantaneous attention this criterion is equivalent to partial balance.

The theory of insensitivity within generalised semi-Markov processes is extended to cover classes of models in which the generally distributed lifetimes can be terminated prematurely by the deaths of negative exponentially distributed lifetimes. As a consequence of this approach it is shown that there exist classes of processes which are insensitive with respect to characteristics of the general distributions other than the mean. Two examples are given. The first is an analysis of networks of queues in which the generally distributed service times can be interrupted with resulting changes in routing probabilities. The second is a model for the effect of disturbances on the evolution of a vegetation community.

The theory of insensitivity is extended to include processes where some of the routing probabilities of the process are made dependent upon the length of time the generally distributed variables have been alive. Examples of such systems are given, including networks of queues with age-dependent routing.

This paper generalizes previous results for sojourn-time distributions along so-called overtake-free routes in product-form networks of queues.

This paper contains two contributions to the insensitivity theory for generalized semi-Markov schemes, namely

1. (1) a detailed account of a close relationship between insensitive schemes and partially balanced discrete-state Markov chains, and

2. (2) an extension of the notion of an insensitive scheme in order to incorporate insensitivity phenomena not previously covered.

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.